Allometric scaling, size distribution and pattern formation of natural cities

There has been a strong interest in more cogent definitions on economies of scale to reveal general urban growth laws and to develop urban performance metrics. Unstructured data, including satellite images, will provide us with new sources to do so by defining cities as aggregates of human activities. Such a uniform definition on the basis of nighttime light clusters is more consistent and economically meaningful than administrative or official boundaries. In comparison with patterns of city systems found in traditional census data, we examined the allometric scaling, size distribution and fractal geometry of natural cities. It can be concluded from our empirical analysis on regional, country and continental scales that a super-linear scaling between lightness and area with a stable exponent across different low light threshold levels generally holds for natural cities. But Zipf's Law does not always apply over the whole range of lightness thresholds. Furthermore, we build a model based on the simple geometric matching mechanism to reproduce the self-organized formation process of nighttime light patterns. The statistical properties including allometries, size distributions and fractal geometries generated by our model are in good agreement with empirical evidence. These findings have profound implications for understanding the effects of simple aggregation behaviour in primitive stages of city formation and the urbanization process.